Mathematical Definitions and Derivations

Intermodulation Calculations

Non-linearities in components create unwanted or spurious products from
multiple tone inputs. Of particular concern in communications are the
2nd and 3rd order products. The relationships between fundamental power,
spurious level and intercept point of these products use the following
definitions:

Note — replace n
with 2 or 3 when referring to 2nd or 3rd order calculations respectively

the fundamental input power (dBm)

the fundamental output power (dBm)

the nth order input intercept point (dBm)

the spurious level of the nth order input
intermodulation product (dBm)

the nth order output intercept point (dBm)

the spurious level of the nth order output intermodulation product (dBm)

the nth order output rejection ratio or relative level (dBc) of the
spurious products compared to the fundamental signals.

the nth order input rejection ratio. This is the amount the input level
must be increased (dB) to raise the nth order spurious products to the
same level as the input before the increase.

2nd
Order Intermodulation Calculations

The equivalent 2nd order output intercept
point for two cascaded stages is given by

The input intercept point of the cascaded
stages is related to the output intercept point by the total gain:

Filtering Effects on IPx

Normal two-tone intermodulation performance
assumes that

the interfering tones are spaced at equal intervals
from the desired signal, and

the interfering tones are in the signal passband.

Filtering is often introduced to the system
to reduce the interfering tones relative to the desired signal. Because
of the filtering, input levels to subsequent devices are reduced, and,
consequently, produce less intermodulation distortion in the final output.
The
net effect is to create an apparent IPx that is greater than the inherent
IPx of the system without the filters. The calculations
that are normally used in calculating the system intermodulation performance
must be modified to accommodate the effects of filtering as

Effective 2nd order intercept

Effective 3rd order intercept

where,

X = the relative attenuation of the adjacent
interferer

Y = the relative attenuation of the alternate
interferer.

Note 1 Cascaded intermodulation calculations compute the worst-case
scenario by summing the products constructively from stage to stage. To
combine in any other manner would require knowledge of the transmission
phase characteristics of the components.

Noise Calculations

The NF (noise figure) of a system is a measure of SNR (signal-to-noise
ratio) degradation as a signal passes through the system. For a constant
bandwidth, the SNR at the output will always be less than the ratio at
the input due to the added noise by the system. The degree of degradation
depends on the equivalent noise temperature of the system and the noise
temperature of the source. For instance, if the noise added by the system
has the same power as the source noise then the composite noise will be
3 dB higher, and the output SNR 3 dB lower. This relationship is expressed
in the general equation

where

is the equivalent noise temperature of the system
( Kelvin)

is the source noise temperature ( Kelvin)

The standard definition of NF is based on
a source temperature of 290 Kelvin; hence, the common equation

In practice, however, the source temperature may be quite low as when
an antenna is pointed into deep space (20 K); therefore
a receiver will have a different actual
noise figure when the antenna is pointed at deep space than when pointed
at a tower on the horizon. The receiver noise temperature hasn't changed
only the source to which it is compared. The general
equation above computes the actual noise figure. The decibel version is

When two or more stages are cascaded the equivalent noise temperature
( Kelvin) can be computed as